Car- is the Sindarin verb for “do, make”, derived from the root √KAR of the same meaning. Tolkien often used it for examples of verb inflections in his writings.
Conceptual Development: The earliest iteration of this verb was G. {car- >>} cartha- “make, finish” in the Gnomish Lexicon of the 1910s with past tense côri (GL/25), reappearing as cartha- “to finish” in the Gnomish Lexicon Slips modifying that document with past tense cair- or cawr- (PE13/111). A set of rough verb forms caron, {cur}, côr, {carn}, crantha, cor and cranthi appear in the margin of a page of the Early Noldorin Grammar, which the editors suggested might be related to the (unglossed and rejected) sentence ᴱN. on gós i·bhelon ar cranthi gwaist ’worin o nomad othra, perhaps meaning “✱he made everyone aware of your sinking” (PE13/128 and note #76).
In the Early Noldorin Dictionary, also from the 1920s, Tolkien had ᴱN. cara- “to make, do, perform, act (trans. and absolute)” with a new past form agor (PE13/161). Hints of the verb N. car- can be found in The Etymologies of the 1930s in words like N. ceredir “doer, maker” and N. osgar “cut round, amputate” (Ety/DER, OS). The verb appeared regularly in conjugation charts and sentences Tolkien’s later writings of the 1950s and 60s (VT50/22; PE17/132).
The root for “head” was established very early in Tolkien’s Elvish languages, appearing in the Qenya Lexicon of the 1910s as ᴱ√KASA “head” (QL/45), though in this period its Qenya derivative was ᴱQ. kar (kas-) because [[eq|final [s] became [r]]] in Early Quenya (PE12/26). It had derivatives in the contemporaneous Gnomish Lexicon such as G. cas “head, skull” (GL/25), a word that reappeared in Early Noldorin word lists of the 1920s as ᴱN. cas “skull” (PE13/140).
The root ᴹ√KAS “head” reappeared in The Etymologies of the 1930s, still with the form ᴹQ. kár (kas-) “head” (Ety/KAS), but Tolkien eventually abandoned the Early Qenya phonology and the Quenya form became Q. kas after some vacillation (PE19/103). The root √KAS “head” continued to appear frequently in Tolkien’s later writing (PE17/114; PE21/70; VT42/12).